Sequence Comparison.pdf

6.4 Asymptotic Analysis of Hit Probability 105
g(x)
0.875
x
Fig. 6.1 The graph of g(x)=x w (1 − x) − p w w
q when w=7 and p=0.7. It increases in (0, w+1
) and
decreases in (
w+1 w ,1).
where the equality sign is possible only if all terms on the left have the same argument,
that is, if r = r 0 . Hence, r 0 is larger in absolute value than any other root of
f (x).
Let f (x) has the following distinct roots
r 0 ,r 1 ,r 2 ,···,r wθ −1,
where r 0 > |r 1 |≥|r 2 |≥···≥|r wθ −1|. Then, By (6.10), we have that
¯Θ n = a 0 r n 0 + a 1r n 1 + ···+ a w θ −1r n w θ −1 . (6.11)
where a i s are constants to be determined. Because
θ i = ¯Θ i−1 − ¯Θ i = p w q
for any i = w θ + 1,...,2w θ , we obtain the following linear equation system with
a i ’s as variables
⎧
⎪⎨
⎪⎩
a 0 (1 − r 0 )r w θ
0
+ a 1 (1 − r 1 )r w θ
1
+ ··· + a wθ −1(1 − r wθ −1)r w θ
w θ −1
a 0 (1 − r 0 )r w θ +1
0
+ a 1 (1 − r 1 )r w θ +1
1
+ ··· + a wθ −1(1 − r wθ −1)r w θ +1
w θ −1
···
a 0 (1 − r 0 )r 2w θ −1
0
+ a 1 (1 − r 1 )r 2w θ −1
1
+ ··· + a wθ −1(1 − r wθ −1)r 2w θ −1
w θ −1
Solving this linear equation system and using r w θ
i (1 − r i )=p w θ q, we obtain
p w qf(1)
a i =
(1 − r i ) 2 r w θ
i
f ′ (r i ) = (p − r i )r i
q[w θ − (w θ + 1)r i ] , i = 1,2,...,w θ − 1.
Thus, (6.11) implies
= p w q
= p w q
= p w q